International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 2.2, pp. 215243
doi: 10.1107/97809553602060000764 Chapter 2.2. Direct methods ^{a}Dipartimento Geomineralogico, Campus Universitario, 70125 Bari, Italy, and Institute of Crystallography, Via G. Amendola, 122/O, 70125 Bari, Italy Direct methods are essentially reciprocalspace techniques, developed historically to solve smallmolecule crystal structures. Their success in this area (in practice, they solve the phase problem) is based on numerous theoretical achievements which concern origin specification (Section 2.2.3), the concepts of structure invariants and seminvariants, normalization of the structure factors (Section 2.2.4), inequalities among structure factors, and probabilistic phase relationships (Section 2.2.5). Probabilistic phase relationships are at the core of direct methods: triplet (via Cochran and via the P10 formula), quartet (according to Hauptman and to Giacovazzo) and quintet invariant phase estimates are described, along with estimates of onephase and of twophase structure seminvariants (via representation theory). Determinantal formulas are also quoted. Techniques for estimating invariants by using some prior information and their realspace counterparts are discussed. The success of direct methods is intimately connected with the phasing procedures. The most important tools of the procedures (e.g. the tangent formula, magic integers, randomstart approaches and figures of merit for recognizing the correct solution) are analysed. The integration of direct methods with macromolecular crystallography is discussed in Section 2.2.10: in particular we refer to the ab initio methods for solving protein structures (in combination with directspace techniques like electrondensity modification, envelope determination, histogram matching etc.) as well as to the combination of direct methods with SIR–MIR and SAD–MAD techniques (both in `onestep' procedures and in the twostep method, the latter requiring the prior determination of the heavyatom or anomalousscatterer substructure). 
Direct methods are today the most widely used tool for solving small crystal structures. They work well both for equalatom molecules and when a few heavy atoms exist in the structure. In recent years the theoretical background of direct methods has been improved to take into account a large variety of prior information (the form of the molecule, its orientation, a partial structure, the presence of pseudosymmetry or of a superstructure, the availability of isomorphous data or of data affected by anomalousdispersion effects, …). Owing to this progress and to the increasing availability of powerful computers, the phase problem for small molecules has been solved in practice: a number of effective, highly automated packages are today available to the scientific community.
The combination of direct methods with socalled directspace methods have recently allowed the ab initio crystal structure solution of proteins. The present limit of complexity is about 2500 nonhydrogen atoms in the asymmetric unit, but diffraction data at atomic resolution (~1 Å) are required. Trials are under way to bring this limit to 1.5 Å and have shown some success.
The theoretical background and tables useful for origin specification are given in Section 2.2.3; in Section 2.2.4 the procedures for normalizing structure factors are summarized. Phasedetermining formulae (inequalities, probabilistic formulae for triplet, quartet and quintet invariants, and for one and twophase s.s.'s, determinantal formulae) are given in Section 2.2.5. In Section 2.2.6 the connection between direct methods and related techniques in real space is discussed. Practical procedures for solving smallmolecule crystal structures are described in Sections 2.2.7 and 2.2.8, and references to the most extensively used packages are given in Section 2.2.9. The integration of direct methods, isomorphous replacement and anomalousdispersion techniques is briefly discussed in Section 2.2.10.
The reader interested in a more detailed description of the topic is referred to a recent textbook (Giacovazzo, 1998).

The normalized structure factors E (see also Chapter 2.1 ) are calculated according to (Hauptman & Karle, 1953) where is the squared observed structurefactor magnitude on the absolute scale and is the expected value of .
depends on the available a priori information. Often, but not always, this may be considered as a combination of several typical situations. We mention:

When probability theory is not used, the quasinormalized structure factors and the unitary structure factors are often used. and are defined according to Since is the largest possible value for represents the fraction of with respect to its largest possible value. Therefore If atoms are equal, then .
N.s.f.'s cannot be calculated by applying (2.2.4.1) to observed s.f.'s because: (a) the observed magnitudes (already corrected for Lp factor, absorption, …) are on a relative scale; (b) cannot be calculated without having estimated the vibrational motion of the atoms.
This is usually obtained by the well known Wilson plot (Wilson, 1942), according to which observed data are divided into ranges of and averages of intensity are taken in each shell. Reflection multiplicities and other effects of spacegroup symmetry on intensities must be taken into account when such averages are calculated. The shells are symmetrically overlapped in order to reduce statistical fluctuations and are restricted so that the number of reflections in each shell is reasonably large. For each shell should be obtained, where K is the scale factor needed to place Xray intensities on the absolute scale, B is the overall thermal parameter and is the expected value of in which it is assumed that all the atoms are at rest. depends upon the structural information that is available (see Section 2.2.4.1 for some examples).
Equation (2.2.4.3) may be rewritten as which plotted at various should be a straight line of which the slope (2B) and intercept (ln K) on the logarithmic axis can be obtained by applying a linear leastsquares procedure.
Very often molecular geometries produce perceptible departures from linearity in the logarithmic Wilson plot. However, the more extensive the available a priori information on the structure is, the closer, on the average, are the Wilsonplot curves to their leastsquares straight lines.
Accurate estimates of B and K require good strategies (Rogers & Wilson, 1953) for:

Once K and B have been estimated, values can be obtained from experimental data by where is the expected value of for the reflection h on the basis of the available a priori information.
Under some fairly general assumptions (see Chapter 2.1 ) probability distribution functions for the variable for cs. and ncs. structures are (see Fig. 2.2.4.1) andrespectively. Corresponding cumulative functions are (see Fig. 2.2.4.2)
Some moments of the distributions (2.2.4.4) and (2.2.4.5) are listed in Table 2.2.4.1. In the absence of other indications for a given crystal structure, a cs. or an ncs. space group will be preferred according to whether the statistical tests yield values closer to column 2 or to column 3 of Table 2.2.4.1.

For further details about the distribution of intensities see Chapter 2.1 .
From the earliest periods of Xray structure analysis several authors (Ott, 1927; Banerjee, 1933; Avrami, 1938) have tried to determine atomic positions directly from diffraction intensities. Significant developments are the derivation of inequalities and the introduction of probabilistic techniques via the use of joint probability distribution methods (Hauptman & Karle, 1953).
An extensive system of inequalities exists for the coefficients of a Fourier series which represents a positive function. This can restrict the allowed values for the phases of the s.f.'s in terms of measured structurefactor magnitudes. Harker & Kasper (1948) derived two types of inequalities:
Type 1. A modulus is bound by a combination of structure factors: where m is the order of the point group and .
Applied to loworder space groups, (2.2.5.1) gives The meaning of each inequality is easily understandable: in , for example, must be positive if is large enough.
Type 2. The modulus of the sum or of the difference of two structure factors is bound by a combination of structure factors: where stands for `real part of'. Equation (2.2.5.2) applied to P1 gives
A variant of (2.2.5.2) valid for cs. space groups is After Harker & Kasper's contributions, several other inequalities were discovered (Gillis, 1948; Goedkoop, 1950; Okaya & Nitta, 1952; de Wolff & Bouman, 1954; Bouman, 1956; Oda et al., 1961). The most general are the Karle–Hauptman inequalities (Karle & Hauptman, 1950): The determinant can be of any order but the leading column (or row) must consist of U's with different indices, although, within the column, symmetryrelated U's may occur. For and , equation (2.2.5.3) reduces to which, for cs. structures, gives the Harker & Kasper inequality For , equation (2.2.5.3) becomes from which where If the moduli , , are large enough, (2.2.5.4) is not satisfied for all values of . In cs. structures the eventual check that one of the two values of does not satisfy (2.2.5.4) brings about the unambiguous identification of the sign of the product .
It was observed (Gillis, 1948) that `there was a number of cases in which both signs satisfied the inequality, one of them by a comfortable margin and the other by only a relatively small margin. In almost all such cases it was the former sign which was the correct one. That suggests that the method may have some power in reserve in the sense that there are still fundamentally stronger inequalities to be discovered'. Today we identify this power in reserve in the use of probability theory.
For any space group (see Section 2.2.3) there are linear combinations of phases with cosines that are, in principle, fixed by the magnitudes alone (s.i.'s) or by the values and the trigonometric form of the structure factor (s.s.'s). This result greatly stimulated the calculation of conditional distribution functions where , is an s.i. or an s.s. and is a suitable set of diffraction magnitudes. The method was first proposed by Hauptman & Karle (1953) and was developed further by several authors (Bertaut, 1955a,b, 1960; Klug, 1958; Naya et al., 1964, 1965; Giacovazzo, 1980a). From a probabilistic point of view the crystallographic problem is clear: the joint distribution , from which the conditional distributions (2.2.5.5) can be derived, involves a number of normalized structure factors each of which is a linear sum of random variables (the atomic contributions to the structure factors). So, for the probabilistic interpretation of the phase problem, the atomic positions and the reciprocal vectors may be considered as random variables. A further problem is that of identifying, for a given Φ, a suitable set of magnitudes on which Φ primarily depends. The formulation of the nested neighbourhood principle first (Hauptman, 1975) fixed the idea of defining a sequence of sets of reflections each contained in the succeeding one and having the property that any s.i. or s.s. may be estimated via the magnitudes constituting the various neighbourhoods. A subsequent more general theory, the representation method (Giacovazzo, 1977a, 1980b), arranges for any Φ the set of intensities in a sequence of subsets in order of their expected effectiveness (in the statistical sense) for the estimation of Φ.
In the following sections the main formulae estimating loworder invariants and seminvariants or relating phases to other phases and diffraction magnitudes are given.
The basic formula for the estimation of the triplet phase given the parameter is Cochran's (1955) formula where , is the atomic number of the jth atom and is the modified Bessel function of order n. In Fig. 2.2.5.1 the distribution is shown for different values of G.
The conditional probability distribution for , given a set of and , is given (Karle & Hauptman, 1956; Karle & Karle, 1966) bywhere is the most probable value for . The variance of may be obtained from (2.2.5.7) and is given by which is plotted in Fig. 2.2.5.2.
Equation (2.2.5.9) is the socalled tangent formula. According to (2.2.5.10), the larger is α the more reliable is the relation .
For an equalatom structure .
The basic conditional formula for sign determination of in cs. crystals is Cochran & Woolfson's (1955) formula where is the probability that is positive and k ranges over the set of known values . The larger the absolute value of the argument of tanh, the more reliable is the phase indication.
An auxiliary formula exploiting all the 's in reciprocal space in order to estimate a single Φ is the formula (Hauptman & Karle, 1958; Karle & Hauptman, 1958) given by where C is a constant which differs for cs. and ncs. crystals, is the average value of and p is normally chosen to be some small number. Several modifications of (2.2.5.12) have been proposed (Hauptman, 1964, 1970; Karle, 1970a; Giacovazzo, 1977b).
A recent formula (Cascarano, Giacovazzo, Camalli et al., 1984) exploits information contained within the second representation of Φ, that is to say, within the collection of special quintets (see Section 2.2.5.6): where k is a free vector. The formula retains the same algebraic form as (2.2.5.6), but where , is assumed to be zero if it is experimentally negative. The prime to the summation warns the reader that precautions have to be taken in order to avoid duplications in the contributions.
G may be positive or negative. In particular, if the triplet is estimated negative.
The accuracy with which the value of Φ is estimated strongly depends on . Thus, in practice, only a subset of reciprocal space (the reflections k with large values of ) may be used for estimating Φ.
(2.2.5.13) proved to be quite useful in practice. Positive triplet cosines are ranked in order of reliability by (2.2.5.13) markedly better than by Cochran's parameters. Negative estimated triplet cosines may be excluded from the phasing process and may be used as a figure of merit for finding the correct solution in a multisolution procedure.
A strength of direct methods is that no knowledge of structure is required for their application. However, when some a priori information is available, it should certainly be a weakness of the methods not to make use of this knowledge. The conditional distribution of Φ given and the first three of the five kinds of a priori information described in Section 2.2.4.1 is (Main, 1976; Heinermann, 1977a) where stand for h, , , and for . The quantities have been calculated in Section 2.2.4.1 according to different categories: is a suitable average of the product of three scattering factors for the ith atomic group, p is the number of atomic groups in the cell including those related by symmetry elements. We have the following categories.

In early papers (Hauptman & Karle, 1953; Simerska, 1956) the phase was always expected to be zero. Schenk (1973a,b) [see also Hauptman (1974)] suggested that Φ primarily depends on the seven magnitudes: , called basis magnitudes, and , called cross magnitudes.
The conditional probability of Φ in P1 given seven magnitudes according to Hauptman (1975) is where L is a suitable normalizing constant which can be derived numerically, For equal atoms . Denoting gives Fig. 2.2.5.3 shows the distribution (2.2.5.18) for three typical cases. It is clear from the figure that the cosine estimated near π or in the middle range will be in poorer agreement with the true values than the cosine near 0 because of the relatively larger values of the variance. In principle, however, the formula is able to estimate negative or enantiomorphsensitive quartet cosines from the seven magnitudes.

Distributions (2.2.5.18) (solid curve) and (2.2.5.20) (dashed curve) for the indicated values in three typical cases. 
In the cs. case (2.2.5.18) is replaced (Hauptman & Green, 1976) by where is the probability that the sign of is positive or negative, and The normalized probability may be derived by . More simple probabilistic formulae were derived independently by Giacovazzo (1975, 1976): where and . Q is never allowed to be negative.
According to (2.2.5.20) is expected to be positive or negative according to whether is positive or negative: the larger is C, the more reliable is the phase indication. For , (2.2.5.18) and (2.2.5.20) are practically equivalent in all cases. If N is small, (2.2.5.20) is in good agreement with (2.2.5.18) for quartets strongly defined as positive or negative, but in poor agreement for enantiomorphsensitive quartets (see Fig. 2.2.5.3).
In cs. cases the sign probability for is where G is defined by (2.2.5.21).
All three cross magnitudes are not always in the set of measured reflections. From marginal distributions the following formulae arise (Giacovazzo, 1977c; Heinermann, 1977b):
Equations (2.2.5.20) and (2.2.5.23) are easily modifiable when some cross magnitudes are not in the measurements. If is not measured then (2.2.5.20) or (2.2.5.23) are still valid provided that in G it is assumed that . For example, if and are not in the data then (2.2.5.21) and (2.2.5.22) become In space groups with symmetry higher than more symmetryequivalent quartets can exist of the type where are rotation matrices of the space group. The set is called the first representation of Φ. In this case Φ primarily depends on more than seven magnitudes. For example, let us consider in Pmmm the quartet Quartets symmetry equivalent to Φ and respective cross terms are given in Table 2.2.5.1.

Experimental tests on the application of the representation concept to quartets have been made (Busetta et al., 1980). It was shown that quartets with more than three cross magnitudes are more accurately estimated than other quartets. Also, quartets with a cross reflection which is systematically absent were shown to be of significant importance in direct methods. In this context it is noted that systematically absent reflections are not usually included in the set of diffraction data. This custom, not exceptionable when only triplet relations are used, can give rise to a loss of information when quartets are used. In fact the usual programs of direct methods discard quartets as soon as one of the cross reflections is not measured, so that systematic absences are dealt with in the same manner as those reflections which are outside the sphere of measurements.
A quintet phase may be considered as the sum of three suitable triplets or the sum of a triplet and a quartet, i.e. or It depends primarily on 15 magnitudes: the five basis magnitudesand the ten cross magnitudesIn the following we will denoteConditional distributions of Φ in P1 and given the 15 magnitudes have been derived by several authors and allow in favourable circumstances in ncs. space groups the quintets having Φ near 0 or near π or near to be identified. Among others, we remember:

For cs. cases (2.2.5.24) reduces to Positive or negative quintets may be identified according to whether G is larger or smaller than zero.
If is not measured then (2.2.5.24) and (2.2.5.25) are still valid provided that in (2.2.5.25) .
If the symmetry is higher than in then more symmetryequivalent quintets can exist of the type where are rotation matrices of the space groups. The set is called the first representation of Φ. In this case Φ primarily depends on more than 15 magnitudes which all have to be taken into account for a careful estimation of Φ (Giacovazzo, 1980a).
A wide use of quintet invariants in directmethods procedures is prevented for two reasons: (a) the large correlation of positive quintet cosines with positive triplets; (b) the large computing time necessary for their estimation [quintets are phase relationships of order , so a large number of quintets have to be estimated in order to pick up a sufficient percentage of reliable ones].
In a crystal structure with N identical atoms the joint probability distribution of n normalized s.f.'s under the following conditions:
is given (Tsoucaris, 1970) [see also Castellano et al. (1973) and Heinermann et al. (1979)] by for cs. structures and for ncs. structures. In (2.2.5.27) and (2.2.5.28) we have denoted is an element of , and is the covariance matrix with elements is a K–H determinant: therefore . Let us call the K–H determinant obtained by adding to the last column and line formed by , and , respectively. Then (2.2.5.27) and (2.2.5.28) may be written and respectively. Because is a constant, the maximum values of the conditional joint probabilities (2.2.5.29) and (2.2.5.30) are obtained when is a maximum. Thus the maximum determinant rule may be stated (Tsoucaris, 1970; Lajzérowicz & Lajzérowicz, 1966): among all sets of phases which are compatible with the inequality the most probable one is that which leads to a maximum value of .
If only one phase, i.e. , is unknown whereas all other phases and moduli are known then (de Rango et al., 1974; Podjarny et al., 1976) for cs. crystals and for ncs. crystals where Equations (2.2.5.31) and (2.2.5.32) generalize (2.2.5.11) and (2.2.5.7), respectively, and reduce to them for . Fourthorder determinantal formulae estimating triplet invariants in cs. and ncs. crystals, and making use of the entire data set, have recently been secured (Karle, 1979, 1980a).
Advantages, limitations and applications of determinantal formulae can be found in the literature (Heinermann et al., 1979; de Rango et al., 1975, 1985). Taylor et al. (1978) combined K–H determinants with a magicinteger approach. The computing time, however, was larger than that required by standard computing techniques. The use of K–H matrices has been made faster and more effective by de Gelder et al. (1990) (see also de Gelder, 1992). They developed a phasing procedure (CRUNCH) which uses random phases as starting points for the maximization of the K–H determinants.
According to the representations method (Giacovazzo, 1977a, 1980a,b):

The more general expressions for the s.s.'s of first rank are
In other words:

The set of special quartets (2.2.5.35a) and (2.2.5.35b) constitutes the first representations of Φ.
Structure seminvariants of the second rank can be characterized as follows: suppose that, for a given seminvariant Φ, it is not possible to find a vectorial index h and a rotation matrix such that is a structure invariant. Then Φ is a structure seminvariant of the second rank and a set of structure invariants ψ can certainly be formed, of type by means of suitable indices h and l and rotation matrices and . As an example, for symmetry class 222, or or are s.s.'s of the first rank while is an s.s. of the second rank.
The procedure may easily be generalized to s.s.'s of any order of the first and of the second rank. So far only the role of onephase and twophase s.s.'s of the first rank in direct procedures is well documented (see references quoted in Sections 2.2.5.9 and 2.2.5.10).
Let be our onephase s.s. of the first rank, where In general, more than one rotation matrix and more than one vector h are compatible with (2.2.5.36). The set of special triplets is the first representation of . In cs. space groups the probability that , given and the set , may be estimated (Hauptman & Karle, 1953; Naya et al., 1964; Cochran & Woolfson, 1955) by where In (2.2.5.37), the summation over n goes within the set of matrices for which (2.2.5.35a,b) is compatible, and h varies within the set of vectors which satisfy (2.2.5.36) for each . Equation (2.2.5.36) is actually a generalized way of writing the socalled relationships (Hauptman & Karle, 1953).
If is a phase restricted by symmetry to and in an ncs. space group then (Giacovazzo, 1978) If is a general phase then is distributed according to where with a reliability measured by The second representation of is the set of special quintets provided that h and vary over the vectors and matrices for which (2.2.5.36) is compatible, k over the asymmetric region of the reciprocal space, and over the rotation matrices in the space group. Formulae estimating via the second representation in all the space groups [all the base and cross magnitudes of the quintets (2.2.5.40) now constitute the a priori information] have been secured (Giacovazzo, 1978; Cascarano & Giacovazzo, 1983; Cascarano, Giacovazzo, Calabrese et al., 1984). Such formulae contain, besides the contribution of order provided by the first representation, a supplementary (not negligible) contribution of order arising from quintets.
Denoting formulae (2.2.5.37), (2.2.5.38), (2.2.5.39) still hold provided that is replaced by where m is the number of symmetry operators and is the Hermite polynomial of order four.
is assumed to be zero if it is computed negative. The prime to the summation warns the reader that precautions have to be taken in order to avoid duplication in the contributions.
Twophase s.s.'s of the first rank were first evaluated in some cs. space groups by the method of coincidence by Grant et al. (1957); the idea was extended to ncs. space groups by Debaerdemaeker & Woolfson (1972), and in a more general way by Giacovazzo (1977e,f).
The technique was based on the combination of the two triplets which, subtracted from one another, give If all four 's are sufficiently large, an estimate of the twophase seminvariant is available.
Probability distributions valid in according to the neighbourhood principle have been given by Hauptman & Green (1978). Finally, the theory of representations was combined by Giacovazzo (1979a) with the joint probability distribution method in order to estimate twophase s.s.'s in all the space groups.
According to representation theory, the problem is that of evaluating via the special quartets (2.2.5.35a) and (2.2.5.35b). Thus, contributions of order will appear in the probabilistic formulae, which will be functions of the basis and of the cross magnitudes of the quartets (2.2.5.35) . Since more pairs of matrices and can be compatible with (2.2.5.34), and for each pair more pairs of vectors and may satisfy (2.2.5.34), several quartets can in general be exploited for estimating Φ. The simplest case occurs in where the two quartets (2.2.5.35) suggest the calculation of the sixvariate distribution function which leads to the probability formula where is the probability that the product is positive, and It may be seen that in favourable cases .
For the sake of brevity, the probabilistic formulae for the general case are not given and the reader is referred to the original papers.
The statistical treatment suggested by Wilson for scaling observed intensities corresponds, in direct space, to the origin peak of the Patterson function, so it is not surprising that a general correspondence exists between probabilistic formulation in reciprocal space and algebraic properties in direct space.
For a structure containing atoms which are fully resolved from one another, the operation of raising to the nth power retains the condition of resolved atoms but changes the shape of each atom. Let where is an atomic function and is the coordinate of the `centre' of the atom. Then the Fourier transform of the electron density can be written as If the atoms do not overlap and its Fourier transform gives is the scattering factor for the jth peak of :
We now introduce the condition that all atoms are equal, so that and for any j. From (2.2.6.1) and (2.2.6.2) we may write where is a function which corrects for the difference of shape of the atoms with electron distributions and . Since the Fourier transform of both sides gives from which the following relation arises: For , equation (2.2.6.4) reduces to Sayre's (1952) equation [but see also Hughes (1953)] If the structure contains resolved isotropic atoms of two types, P and Q, it is impossible to find a factor such that the relation holds, since this would imply values of such that and simultaneously. However, the following relationship can be stated (Woolfson, 1958): where and are adjustable parameters of . Equation (2.2.6.6) can easily be generalized to the case of structures containing resolved atoms of more than two types (von Eller, 1973).
Besides the algebraic properties of the electron density, Patterson methods also can be developed so that they provide phase indications. For example, it is possible to find the reciprocal counterpart of the function For the function (2.2.6.7) coincides with the usual Patterson function ; for , (2.2.6.7) reduces to the double Patterson function introduced by Sayre (1953). Expansion of as a Fourier series yields Vice versa, the value of a triplet invariant may be considered as the Fourier transform of the double Patterson.
Among the main results relating direct and reciprocalspace properties it may be remembered:

A traditional procedure for phase assignment may be schematically presented as follows:

In very complex structures a large initial set of known phases seems to be a basic requirement for a structure to be determined. This aim can be achieved, for example, by introducing a large number of permutable phases into the initial set. However, the introduction of every new symbol implies a fourfold increase in computing time, which, even in fast computers, quickly leads to computingtime limitations. On the other hand, a relatively large starting set is not in itself enough to ensure a successful structure determination. This is the case, for example, when the triplet invariants used in the initial steps differ significantly from zero. New strategies have therefore been devised to solve more complex structures.

Some references for directmethods packages are given below. Other useful packages using symbolic addition or multisolution procedures do exist but are not well documented.
CRUNCH: Gelder, R. de, de Graaff, R. A. G. & Schenk, H. (1993). Automatic determination of crystal structures using Karle–Hauptman matrices. Acta Cryst. A49, 287–293.
DIRDIF: Beurskens, P. T., Beurskens G., de Gelder, R., GarciaGranda, S., Gould, R. O., Israel, R. & Smits, J. M. M. (1999). The DIRDIF99 program system. Crystallography Laboratory, University of Nijmegen, The Netherlands.
MITHRIL: Gilmore, C. J. (1984). MITHRIL. An integrated directmethods computer program. J. Appl. Cryst. 17, 42–46.
MULTAN88: Main, P., Fiske, S. J., Germain, G., Hull, S. E., Declercq, J.P., Lessinger, L. & Woolfson, M. M. (1999). Crystallographic software: teXsan for Windows. http://www.rigaku.com/downloads/journal/Vol15.1.1998/texsan.pdf .
PATSEE: Egert, E. & Sheldrick, G. M. (1985). Search for a fragment of known geometry by integrated Patterson and direct methods. Acta Cryst. A41, 262–268.
SAPI: Fan, H.F. (1999). Crystallographic software: teXsan for Windows. http://www.rigaku.com/downloads/journal/Vol15.1.1998/texsan.pdf .
SnB: Weeks, C. M. & Miller, R. (1999). The design and implementation of SnB version 2.0. J. Appl. Cryst. 32, 120–124.
SHELX97 and SHELXS: Sheldrick, G. M. (2000). The SHELX home page. http://shelx.uniac.gwdg.de/SHELX/ .
SHELXD: Sheldrick, G. M. (1998). SHELX: applications to macromolecules. In Direct methods for solving macromolecular structures, edited by S. Fortier, pp. 401–411. Dordrecht: Kluwer Academic Publishers.
SIR97: Altomare, A., Burla, M. C., Camalli, M., Cascarano, G. L., Giacovazzo, C., Guagliardi, A., Moliterni, A. G. G., Polidori, G. & Spagna, R. (1999). SIR97: a new tool for crystal structure determination and refinement. J. Appl. Cryst. 32, 115–119.
SIR2004: Burla, M. C., Caliandro, R., Camalli, M., Carrozzini, B., Cascarano, G. L., De Caro, L., Giacovazzo, C., Polidori, G. & Spagna, R. (2005). SIR2004: an improved tool for crystal structure determination and refinement. J. Appl. Cryst. 38, 381–388.
XTAL3.6.1: Hall, S. R., du Boulay, D. J. & OlthofHazekamp, R. (1999). Xtal3.6 crystallographic software. http://xtal.sourceforge.net/ .
The smallest protein molecules contain about 400 nonhydrogen atoms, so they cannot be solved ab initio by the algorithms specified in Sections 2.2.7 and 2.2.8. However, traditional direct methods are applied for:
The application of standard tangent techniques to (a) and (b) has not been found to be very satisfactory (Coulter & Dewar, 1971; Hendrickson et al., 1973; Weinzierl et al., 1969). Tangent methods, in fact, require atomicity and nonnegativity of the electron density. Both these properties are not satisfied if data do not extend to atomic resolution (d > 1.2 Å). Because of series termination and other errors the electrondensity map at d > 1.2 Å presents large negative regions which will appear as false peaks in the squared structure. However, tangent methods use only a part of the information given by the Sayre equation (2.2.6.5). In fact, (2.2.6.5) express two equations relating the radial and angular parts of the two sides, so obtaining a large degree of overdetermination of the phases. To achieve this Sayre (1972) [see also Sayre & Toupin (1975)] suggested minimizing (2.2.10.1) by least squares as a function of the phases: Even if tests on rubredoxin (extensions of phases from 2.5 to 1.5 Å resolution) and insulin (Cutfield et al., 1975) (from 1.9 to 1.5 Å resolution) were successful, the limitations of the method are its high cost and, especially, the higher efficiency of the leastsquares method. Equivalent considerations hold for the application of determinantal methods to proteins [see Podjarny et al. (1981); de Rango et al. (1985) and literature cited therein].
A question now arises: why is the tangent formula unable to solve protein structures? Fan et al. (1991) considered the question from a firstprinciple approach and concluded that:
Sheldrick (1990) suggested that direct methods are not expected to succeed if fewer than half of the reflections in the range 1.1–1.2 Å are observed with (a condition seldom satisfied by protein data).
The most complete analysis of the problem has been made by Giacovazzo, Guagliardi et al. (1994). They observed that the expected value of α (see Section 2.2.7) suggested by the tangent formula for proteins is comparable with the variance of the α parameter. In other words, for proteins the signal determining the phase is comparable with the noise, and therefore the phase indication is expected to be unreliable.
Quite relevant results have recently been obtained by integrating direct methods with some additional experimental information. In particular, we will describe the combination of direct methods with:
Point (d) will not be treated here, as it is described extensively in IT F, Part 13 .
Ab initio techniques do not require prior information of any atomic positions. The recent tremendous increase in computing speed led to direct methods evolving towards the rapid development of multisolution techniques. The new algorithms of the program ShakeandBake (Weeks et al., 1994; Weeks & Miller, 1999; Hauptman et al., 1999) allowed an impressive extension of the structural complexity amenable to direct phasing. In particular we mention: (a) the minimal principle (De Titta et al., 1994), according to which the phase problem is considered as a constrained global optimization problem; (b) the refinement procedure, which alternately uses direct and reciprocalspace techniques; and (c) the parametershift optimization technique (Bhuiya & Stanley, 1963), which aims at reducing the value of the minimal function (Hauptman, 1991; De Titta et al., 1994). An effective variant of Shakeand Bake is SHELXD (Sheldrick, 1998) which cyclically alternates tangent refinement in reciprocal space with peaklist optimisation procedures in real space (Sheldrick & Gould, 1995). Detailed information on these programs is available in IT F (2001), Part 16 .
A different approach is used by ACORN (Foadi et al., 2000), which first locates a small fragment of the molecule (eventually by molecularreplacement techniques) to obtain a useful nonrandom starting set of phases, and then refines them by means of solventflattening techniques.
The program SIR2004 (Burla et al., 2005) uses the tangent formula as well as automatic Patterson techniques to obtain a first imperfect structural model; then directspace techniques are used to refine the model. The Patterson approach is based on the use of the superposition minimum function (Buerger, 1959; Richardson & Jacobson, 1987; Sheldrick, 1992; Pavelcík, 1988; Pavelcík et al., 1992; Burla et al., 2004). It may be worth noting that even this approach is of multisolution type: up to 20 trial solutions are provided by using as pivots the highest maxima in the superposition minimum function.
It is today possible to solve structures up to 2500 nonhydrogen atoms in the asymmetric unit provided data at atomic (about 1 Å) resolution are available. Proteins with data at quasiatomic resolution (say up to 1.5–1.6 Å) can also be solved, but with greater difficulties (Burla et al., 2005). A simple evaluation of the potential of the ab initio techniques suggests that the structural complexity range and the resolution limits amenable to the ab initio approach could be larger in the near future. The approach will profit by general technical advances like the increasing speed of computers and by the greater efficiency of informatic tools (e.g. faster Fouriertransform techniques). It could also profit from new specific crystallographic algorithms (for example, Oszlányi & Süto, 2004). It is of particular interest that extrapolating moduli and phases of nonmeasured reflections beyond the experimental resolution limit makes the ab initio phasing process more efficient, and leads to crystal structure solution even in cases in which the standard programs do not succeed (Caliandro et al., 2005a,b). Moreover, the use of the extrapolated values improves the quality of the final electrondensity maps and makes it easier to recognize the correct one among several trial structures.
SIR–MIR cases are characterized by a situation in which there is one native protein and one or more heavyatom substructures. In this situation the phasing procedure may be a twostep process: in the first stage the heavyatom positions are identified by Patterson techniques (Rossmann, 1961; Okaya et al., 1955) or by direct methods (Mukherjee et al., 1989). In the second step the protein phases are estimated by exploiting the substructure information. Direct methods are able to contribute to both steps (see Sections 2.2.10.5 and 2.2.10.6). In Section 2.2.10.4 we show that direct methods are also able to suggest alternative onestep procedures by estimating structure invariants from isomorphous data.
The theoretical basis was established by Hauptman (1982a): his primary interest was to establish the twophase and threephase structure invariants by exploiting the experimental information provided by isomorphous data. The protein phases could be directly assigned via a tangent procedure.
Let us denote the modulus of the isomorphous difference aswhere the subscripts d and p denote the derivative and the protein, respectively.
Denote also by and atomic scattering factors for the atom labelled j in a pair of isomorphous structures, and let and denote corresponding normalized structure factors. Then where The conditional probability of the twophase structure invariant given and is (Hauptman, 1982a)where Threephase structure invariants were evaluated by considering that eight invariants exist for a given triple of indices h, k, l : So, for the estimation of any , the joint probability distribution has to be studied, from which eight conditional probability densities can be obtained: for .
The analytical expressions of are too intricate and are not given here (the reader is referred to the original paper). We only say that may be positive or negative, so that reliable triplet phase estimates near 0 or near π are possible: the larger , the more reliable the phase estimate.
A useful interpretation of the formulae in terms of experimental parameters was suggested by Fortier et al. (1984): according to them, distributions do not depend, as in the case of the traditional threephase invariants, on the total number of atoms per unit cell but rather on the scattering difference between the native protein and the derivative (that is, on the scattering of the heavy atoms in the derivative).
Hauptman's formulae were generalized by Giacovazzo et al. (1988): the new expressions were able to take into account the resolution effects on distribution parameters. The formulae are completely general and include as special cases native protein and heavyatom isomorphous derivatives as well as Xray and neutron diffraction data. Their complicated algebraic forms are easily reduced to a simple expression in the case of a native protein heavyatom derivative: in particular, the reliability parameter for is where indices p and H warn that parameters have to be calculated over protein atoms and over heavy atoms, respectively, and Δ is a pseudonormalized difference (with respect to the heavyatom structure) between moduli of structure factors.
Equation (2.2.10.2) may be compared with Karle's (1983) algebraic rule: if the sign of is plus then the value of is estimated to be zero; if its sign is minus then the expected value of is close to π. In practice Karle's rule agrees with (2.2.10.2) only if the Cochrantype term in (2.2.10.2) may be neglected. Furthermore, (2.2.10.2) shows that large reliability values do not depend on the triple product of structurefactor differences, but on the triple product of pseudonormalized differences.
A similar mathematical approach has been applied to estimate quartet invariants via isomorphous data. The result may be summarized as follows: a quartet is a phase relationship of order (Giacovazzo & Siliqi, 1996a,b; see also Kyriakidis et al., 1996), with reliability factor equal to where Q_{4} is a suitable normalizing factor.
As previously stressed, equations (2.2.10.2) and (2.2.10.3) are valid if the lack of isomorphism and the errors in the measurements are assumed to be negligible. At first sight this approach seems more appealing than the traditional twostep procedures, however it did not prove to be competitive with them. The main reason is the absence in the Hauptman and Giacovazzo approaches of a probabilistic treatment of the errors: such a treatment, on the contrary, is basic for the traditional SIR–MIR techniques [see Blow & Crick (1959) and Terwilliger & Eisenberg (1987) for two related approaches].
The problem of the errors in the probabilistic scenario defined by the joint probability distribution functions approach has recently been overcome by Giacovazzo et al. (2001). In their probabilistic calculations the following assumptions were made:where j refers to the jth derivative. is the error, which can include model as well as measurement errors.
A more realistic expression for the reliability factor G of triplet invariants is obtained by including the expression (2.2.10.4) in the probabilistic approach. Then the reliability parameter of the triplet invariants is transformed into (Giacovazzo et al., 2001)where .
Equation (2.2.10.5) suggests how the error influences the reliability of the triplet estimate: even quite a small value of may be critical if the scattering power of the heavyatom substructure is a very small percentage of the derivative scattering power.
A onestep procedure has been implemented in a computer program (Giacovazzo et al., 2002): it has been shown that the method is able to derive automatically, from the experimental data and without any user intervention, good quality (i.e. perfectly interpretable) electrondensity maps.
2.2.10.5. SIR–MIR case: the twostep procedure. Finding the heavyatom substructure by direct methods
The first trials for finding the heavyatom substructure were based on the following assumption: the modulus of the isomorphous difference,is assumed at a first approximation as an estimate of the heavyatom structure factor F_{H}. Perutz (1956) approximated F_{H}^{2} with the difference . Blow (1958) and Rossmann (1960) suggested a better approximation: . A deeper analysis was performed by Phillips (1966), Dodson & Vijayan (1971), Blessing & Smith (1999) and GrosseKunstleve & Brunger (1999). The use of direct methods requires the normalization of and application of the tangent formula (Wilson, 1978).
A sounder procedure has been suggested by Giacovazzo et al. (2004): they studied, for the SIR case, the joint probability distribution functionunder the following assumptions:

Let us suppose that the various heavyatom substructures have been determined. They may be used as additional prior information for a more accurate estimate of the values. To this purpose the distributionsmay be used under the assumption (2.2.10.6). and , for j = 1, …, n, are the structure factors of the jth derivative and of the jth heavyatom substructure, respectively, both normalized with respect to the protein. Any joint probability density (2.2.10.8) may be reliably approximated by a multidimensional Gaussian distribution (Giacovazzo & Siliqi, 2002), from which the following conditional distribution is obtained:where , the expected value of , is given byand .
is the reliability factor of the phase estimate. A robust phasing procedure has been established which, starting from the observed moduli , is able to automatically provide, without any user intervention, a highquality electrondensity map of the protein (Giacovazzo et al., 2002).
If the frequency of the radiation is close to an absorption edge of an atom, then that atom will scatter the Xrays anomalously (see Chapter 2.4 ) according to . This results in the breakdown of Friedel's law. It was soon realized that the Bijvoet difference could also be used in the determination of phases (Peerdeman & Bijvoet, 1956; Ramachandran & Raman, 1956; Okaya & Pepinsky, 1956). Since then, a great deal of work has been done both from algebraic (see Chapter 2.4 ) and from probabilistic points of view. In this section we are only interested in the second.
SAD (single anomalous dispersion) and MAD (multiple anomalous dispersion) techniques can be used. Both are characterized by one protein structure and one anomalousscatterer substructure. The experimental diffraction data differ only because of the different anomalous scattering (not because of different anomalousscatterer substructures). In the MAD case the anomalousscatterer substructure is in some way `overdetermined' by the data and, therefore, it is more convenient to use a twostep procedure: first define the positions of the anomalous scatterers, and then estimate the protein phase values. For completeness, we describe the onestep procedures in Section 2.2.10.8. These are based on the estimation of the structure invariants and on the application of the tangent formula. The twostep procedures are described in the Sections 2.2.10.9 and 2.2.10.10.
Probability distributions of diffraction intensities and of selected functions of diffraction intensities for dispersive structures have been given by various authors [Parthasarathy & Srinivasan (1964), see also Srinivasan & Parthasarathy (1976) and relevant literature cited therein]. We describe here some probabilistic formulae for estimating invariants of low order.

GivenHauptman and Giacovazzo found the following conditional distribution:
The definitions of Ω and ω are rather extensive and so the reader is referred to the published papers. We only add that Ω is always positive and that ω, the expected value of Φ, may lie anywhere between 0 and 2π. Understanding the role of the various parameters in equation (2.2.10.11) is not easy. Giacovazzo et al. (2003) found an equivalent simpler expression from which interpretable estimates of the parameters were obtained. In the same paper the limitations of the approach (versus the twostep procedures) were clarified.
2.2.10.9. SAD–MAD case: the twostep procedures. Finding the anomalousscatterer substructure by direct methods
The anomalousscatterer substructure is traditionally determined by the techniques suggested by Karle and Hendrickson (Karle, 1980b; Hendrickson, 1985; Pähler et al., 1990; Terwilliger, 1994). The introduction of selenium into proteins as selenomethionine encouraged the secondgeneration direct methods programs [Shake and Bake by Miller et al. (1994); Half bake by Sheldrick (1998); SIR2000N by Burla et al. (2001); ACORN by Foadi et al. (2000)] to locate Se atoms. Since the number of Se atoms may be quite large (up to 200), direct methods rather than Patterson techniques seem to be preferable. Shake and Bake, Half Bake and ACORN obtain the coordinates of the anomalous scatterers from a singlewavelength set of data. When more sets of diffraction data are available the solutions obtained by the other sets are used to confirm the correct solution.
A different approach has been suggested in two recent papers (Burla et al., 2002; Burla, Carrozzini et al., 2003): the estimates of the amplitudes of the structure factors of the anomalously scattering substructure are derived, via the rigorous method of the joint probability distribution functions, from the experimental diffraction moduli relative to n wavelengths. To do that, first the joint distributionis calculated, where A_{oa}, B_{oa}, E_{oa}, , , , are the real and imaginary components of E_{oa}, , , respectively, K is a symmetric square matrix of order (4n + 2), K^{−1} = {λ_{ij}} is its inverse, and T is a suitable vector with components defined in terms of the variables . E_{oa} is the normalized structure factor of the anomalous scatterer substructure calculated by neglecting anomalous scattering components. Then the conditional distributionis derived, from whichis obtained, whereThe standard deviation of the estimate is also calculated:from which
The advantage of the above approach is that the estimates can simultaneously exploit both the anomalous and the dispersive differences. The computing procedure proposed by Burla, Carrozzini et al. (2003) is the following:

The application of the above procedure to several MAD cases showed that the various wavelength combinations are not equally informative. A criterion based on the correlation among the various Δ_{ano} values was also provided (see also Schneider & Sheldrick, 2002) for predicting the most informative combinations.
Once the anomalousscatterer substructure has been found, the corresponding structure factors are known in modulus and phase. Then the conditional joint probability distributionmay be calculated (Giacovazzo & Siliqi, 2004), from which the conditional distributionmay be derived.
It has been shown that the most probable phase of , say , is the phase of the vectorand the reliability parameter of the phase estimate is nothing other than the modulus of (2.2.10.14). The first term in (2.2.10.14) is a Simlike contribution; the other terms, through the weights w, take into account the errors and the experimental differences , and .
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